Limit theorem on convergence to the local time of a Brownian excursion

Authors

  • Valeriy I. Afanasyev Steklov Mathematical Institute, Russian Academy of Sciences, 8 Gubkina Street, Moscow 119333, Russia

Keywords:

random walk, conditional Brownian motion, local time of conditional Brownian motions, functional limit theorem

Abstract

An integer random walk {Si, i ≥ 0} with zero drift and finite variance σ2 is investigated. For a random pro-cess that assigns, to a variable u > 0, the number of hits of the specified random walk into the state \[\left\lfloor {u\sigma \sqrt n } \right\rfloor \] up to time n and is considered under the condition that S1 > 0, …, Sn – 1 > 0, Sn ≤ 0, a functional limit theorem concerning convergence in distribution of the process to the local time of a Brownian excursion is proved.

Author Biography

  • Valeriy I. Afanasyev, Steklov Mathematical Institute, Russian Academy of Sciences, 8 Gubkina Street, Moscow 119333, Russia

    doctor of science (physics and mathematics); leading researcher at the department of discrete mathematics

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Published

2026-05-21

Issue

No. 1 (2026): Journal of the Belarusian State University. Mathematics and Informatics

Section

Probability Theory and Mathematical Statistics

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Copyright (c) 2026 Journal of the Belarusian State University. Mathematics and Informatics

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How to Cite

[1]
Afanasyev, V.I. 2026. Limit theorem on convergence to the local time of a Brownian excursion. Journal of the Belarusian State University. Mathematics and Informatics. 1 (May 2026), 36–52.